Question: Simplify; express your answer in exponential form. Assume $p\neq 0, k\neq 0$. $\dfrac{{p^{-2}}}{{(p^{4}k^{-2})^{-1}}}$
Solution: To start, try working on the numerator and the denominator independently. In the numerator, we have ${p^{-2}}$ to the exponent ${1}$ . Now ${-2 \times 1 = -2}$ , so ${p^{-2} = p^{-2}}$ In the denominator, we can use the distributive property of exponents. ${(p^{4}k^{-2})^{-1} = (p^{4})^{-1}(k^{-2})^{-1}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{p^{-2}}}{{(p^{4}k^{-2})^{-1}}} = \dfrac{{p^{-2}}}{{p^{-4}k^{2}}}$ Break up the equation by variable and simplify. $\dfrac{{p^{-2}}}{{p^{-4}k^{2}}} = \dfrac{{p^{-2}}}{{p^{-4}}} \cdot \dfrac{{1}}{{k^{2}}} = p^{{-2} - {(-4)}} \cdot k^{- {2}} = p^{2}k^{-2}$.